SlideShare ist ein Scribd-Unternehmen logo
1 von 71
Downloaden Sie, um offline zu lesen
Binary Search Trees
CS 302 – Data Structures
Chapter 8
What is a binary tree?
• Property 1: each node can have up to two
successor nodes.
• Property 2: a unique path exists from the
root to every other node
What is a binary tree? (cont.)
Not a valid binary tree!
Some terminology
• The successor nodes of a node are called its children
• The predecessor node of a node is called its parent
• The "beginning" node is called the root (has no parent)
• A node without children is called a leaf
Some terminology (cont’d)
• Nodes are organize in levels (indexed from 0).
• Level (or depth) of a node: number of edges in the path
from the root to that node.
• Height of a tree h: #levels = L
(Warning: some books define h
as #levels-1).
• Full tree: every node has exactly
two children and all the
leaves are on the same level.
not full!
What is the max #nodes
at some level l?
The max #nodes at level is
l 2l
0
2

1
2

2
2

3
2

where l=0,1,2, ...,L-1
What is the total #nodes N
of a full tree with height h?
0 1 1
1
0 1 1 1
1
0
2 2 ... 2 2 1
...
n
h h
n
n i x
x
i
N
x x x x


 


     
    

using the geometric series:
l=0 l=1 l=h-1
What is the height h
of a full tree with N nodes?
2 1
2 1
log( 1) (log )
h
h
N
N
h N O N
 
  
   
Why is h important?
• Tree operations (e.g., insert, delete, retrieve
etc.) are typically expressed in terms of h.
• So, h determines running time!
•What is the max height of a tree with
N nodes?
•What is the min height of a tree with
N nodes?
N (same as a linked list)
log(N+1)
How to search a binary tree?
(1) Start at the root
(2) Search the tree level
by level, until you find
the element you are
searching for or you reach
a leaf.
Is this better than searching a linked list?
No  O(N)
Binary Search Trees (BSTs)
• Binary Search Tree Property:
The value stored at
a node is greater than
the value stored at its
left child and less than
the value stored at its
right child
In a BST, the value
stored at the root of
a subtree is greater
than any value in its
left subtree and less
than any value in its
right subtree!
Binary Search Trees (BSTs)
Where is the
smallest element?
Ans: leftmost element
Where is the largest
element?
Ans: rightmost element
Binary Search Trees (BSTs)
How to search a binary search tree?
(1) Start at the root
(2) Compare the value of
the item you are
searching for with
the value stored at
the root
(3) If the values are
equal, then item
found; otherwise, if it
is a leaf node, then
not found
How to search a binary search tree?
(4) If it is less than the value
stored at the root, then
search the left subtree
(5) If it is greater than the
value stored at the root,
then search the right
subtree
(6) Repeat steps 2-6 for the
root of the subtree chosen
in the previous step 4 or 5
How to search a binary search tree?
Is this better than searching
a linked list?
Yes !! ---> O(logN)
Tree node structure
template<class ItemType>
struct TreeNode<ItemType> {
ItemType info;
TreeNode<ItemType>* left;
TreeNode<ItemType>* right;
};
Binary Search Tree Specification
#include <fstream.h>
struct TreeNode<ItemType>;
enum OrderType {PRE_ORDER, IN_ORDER, POST_ORDER};
template<class ItemType>
class TreeType {
public:
TreeType();
~TreeType();
TreeType(const TreeType<ItemType>&);
void operator=(const TreeType<ItemType>&);
void MakeEmpty();
bool IsEmpty() const;
bool IsFull() const;
int NumberOfNodes() const;
void RetrieveItem(ItemType&, bool& found);
void InsertItem(ItemType);
void DeleteItem(ItemType);
void ResetTree(OrderType);
void GetNextItem(ItemType&, OrderType, bool&);
void PrintTree(ofstream&) const;
private:
TreeNode<ItemType>* root;
};
Binary Search Tree Specification
(cont.)
Function NumberOfNodes
• Recursive implementation
#nodes in a tree =
#nodes in left subtree + #nodes in right subtree + 1
• What is the size factor?
Number of nodes in the tree we are examining
• What is the base case?
The tree is empty
• What is the general case?
CountNodes(Left(tree)) + CountNodes(Right(tree)) + 1
template<class ItemType>
int TreeType<ItemType>::NumberOfNodes() const
{
return CountNodes(root);
}
template<class ItemType>
int CountNodes(TreeNode<ItemType>* tree)
{
if (tree == NULL)
return 0;
else
return CountNodes(tree->left) + CountNodes(tree->right) + 1;
}
Function NumberOfNodes (cont.)
O(N)
Running Time?
Function RetrieveItem
Function RetrieveItem
• What is the size of the problem?
Number of nodes in the tree we are examining
• What is the base case(s)?
1) When the key is found
2) The tree is empty (key was not found)
• What is the general case?
Search in the left or right subtrees
template <class ItemType>
void TreeType<ItemType>:: RetrieveItem(ItemType& item, bool& found)
{
Retrieve(root, item, found);
}
template<class ItemType>
void Retrieve(TreeNode<ItemType>* tree, ItemType& item, bool& found)
{
if (tree == NULL) // base case 2
found = false;
else if(item < tree->info)
Retrieve(tree->left, item, found);
else if(item > tree->info)
Retrieve(tree->right, item, found);
else { // base case 1
item = tree->info;
found = true;
}
}
Function RetrieveItem (cont.)
O(h)
Running Time?
Function
InsertItem
• Use the
binary search
tree property
to insert the
new item at
the correct
place
Function
InsertItem
(cont.)
• Implementing
insertion
resursively
e.g., insert 11
• What is the size of the problem?
Number of nodes in the tree we are examining
• What is the base case(s)?
The tree is empty
• What is the general case?
Choose the left or right subtree
Function InsertItem (cont.)
template<class ItemType>
void TreeType<ItemType>::InsertItem(ItemType item)
{
Insert(root, item);
}
template<class ItemType>
void Insert(TreeNode<ItemType>*& tree, ItemType item)
{
if(tree == NULL) { // base case
tree = new TreeNode<ItemType>;
tree->right = NULL;
tree->left = NULL;
tree->info = item;
}
else if(item < tree->info)
Insert(tree->left, item);
else
Insert(tree->right, item);
}
Function InsertItem (cont.)
O(h)
Running Time?
Function InsertItem (cont.)
Insert 11
• Yes, certain orders might produce very
unbalanced trees!
Does the order of inserting
elements into a tree matter?
Does the
order of
inserting
elements
into a tree
matter?
(cont.)
• Unbalanced trees are not desirable because
search time increases!
• Advanced tree structures, such as red-black
trees, guarantee balanced trees.
Does the order of inserting elements
into a tree matter? (cont’d)
Function DeleteItem
• First, find the item; then, delete it
• Binary search tree property must be
preserved!!
• We need to consider three different cases:
(1) Deleting a leaf
(2) Deleting a node with only one child
(3) Deleting a node with two children
(1) Deleting a leaf
(2) Deleting a node with
only one child
(3) Deleting a node with two
children
• Find predecessor (i.e., rightmost node in the
left subtree)
• Replace the data of the node to be deleted
with predecessor's data
• Delete predecessor node
(3) Deleting a node with two
children (cont.)
• What is the size of the problem?
Number of nodes in the tree we are examining
• What is the base case(s)?
Key to be deleted was found
• What is the general case?
Choose the left or right subtree
Function DeleteItem (cont.)
template<class ItemType>
void TreeType<ItmeType>::DeleteItem(ItemType item)
{
Delete(root, item);
}
template<class ItemType>
void Delete(TreeNode<ItemType>*& tree, ItemType item)
{
if(item < tree->info)
Delete(tree->left, item);
else if(item > tree->info)
Delete(tree->right, item);
else
DeleteNode(tree);
}
Function DeleteItem (cont.)
template <class ItemType>
void DeleteNode(TreeNode<ItemType>*& tree)
{
ItemType item;
TreeNode<ItemType>* tempPtr;
tempPtr = tree;
if(tree->left == NULL) { // right child
tree = tree->right;
delete tempPtr;
}
else if(tree->right == NULL) { // left child
tree = tree->left;
delete tempPtr;
}
else {
GetPredecessor(tree->left, item);
tree->info = item;
Delete(tree->left, item);
}
}
Function DeleteItem (cont.)
2 children
0 children or
1 child
0 children or
1 child
template<class ItemType>
void GetPredecessor(TreeNode<ItemType>* tree, ItemType& item)
{
while(tree->right != NULL)
tree = tree->right;
item = tree->info;
}
Function DeleteItem (cont.)
template<class ItemType>
void TreeType<ItmeType>::DeleteItem(ItemType item)
{
Delete(root, item);
}
template<class ItemType>
void Delete(TreeNode<ItemType>*& tree, ItemType item)
{
if(item < tree->info)
Delete(tree->left, item);
else if(item > tree->info)
Delete(tree->right, item);
else
DeleteNode(tree);
}
Function DeleteItem (cont.)
Running Time?
O(h)
Function DeleteItem (cont.)
Tree Traversals
There are mainly three ways to traverse a tree:
1) Inorder Traversal
2) Postorder Traversal
3) Preorder Traversal
46
Inorder Traversal: A E H J M T Y
‘J’
‘E’
‘A’ ‘H’
‘T’
‘M’ ‘Y’
tree
Visit left subtree first Visit right subtree last
Visit second
Inorder Traversal
• Visit the nodes in the left subtree, then visit
the root of the tree, then visit the nodes in
the right subtree
Inorder(tree)
If tree is not NULL
Inorder(Left(tree))
Visit Info(tree)
Inorder(Right(tree))
Warning: "visit" implies do something with the value at
the node (e.g., print, save, update etc.).
48
Preorder Traversal: J E A H T M Y
‘J’
‘E’
‘A’ ‘H’
‘T’
‘M’ ‘Y’
tree
Visit left subtree second Visit right subtree last
Visit first
Preorder Traversal
• Visit the root of the tree first, then visit the
nodes in the left subtree, then visit the nodes
in the right subtree
Preorder(tree)
If tree is not NULL
Visit Info(tree)
Preorder(Left(tree))
Preorder(Right(tree))
50
‘J’
‘E’
‘A’ ‘H’
‘T’
‘M’ ‘Y’
tree
Visit left subtree first Visit right subtree second
Visit last
Postorder Traversal: A H E M Y T J
Postorder Traversal
• Visit the nodes in the left subtree first, then
visit the nodes in the right subtree, then visit
the root of the tree
Postorder(tree)
If tree is not NULL
Postorder(Left(tree))
Postorder(Right(tree))
Visit Info(tree)
Tree
Traversals:
another
example
Function PrintTree
• We use "inorder" to print out the node values.
• Keys will be printed out in sorted order.
• Hint: binary search could be used for sorting!
A D J M Q R T
Function PrintTree (cont.)
void TreeType::PrintTree(ofstream& outFile)
{
Print(root, outFile);
}
template<class ItemType>
void Print(TreeNode<ItemType>* tree, ofstream& outFile)
{
if(tree != NULL) {
Print(tree->left, outFile);
outFile << tree->info; // “visit”
Print(tree->right, outFile);
}
}
to overload “<<“ or “>>”, see:
http://www.fredosaurus.com/notes-cpp/oop-friends/overload-io.html
Class Constructor
template<class ItemType>
TreeType<ItemType>::TreeType()
{
root = NULL;
}
Class Destructor
How should we
delete the nodes
of a tree?
Use postorder!
Delete the tree in a
"bottom-up"
fashion
Class Destructor (cont’d)
TreeType::~TreeType()
{
Destroy(root);
}
void Destroy(TreeNode<ItemType>*& tree)
{
if(tree != NULL) {
Destroy(tree->left);
Destroy(tree->right);
delete tree; // “visit”
}
}
postorder
Copy Constructor
How should we
create a copy of
a tree?
Use preorder!
Copy Constructor (cont’d)
template<class ItemType>
TreeType<ItemType>::TreeType(const TreeType<ItemType>&
originalTree)
{
CopyTree(root, originalTree.root);
}
template<class ItemType)
void CopyTree(TreeNode<ItemType>*& copy,
TreeNode<ItemType>* originalTree)
{
if(originalTree == NULL)
copy = NULL;
else {
copy = new TreeNode<ItemType>;
copy->info = originalTree->info;
CopyTree(copy->left, originalTree->left);
CopyTree(copy->right, originalTree->right);
}
}
preorder
// “visit”
ResetTree and GetNextItem
• User needs to specify the tree
traversal order.
• For efficiency, ResetTree stores in
a queue the results of the
specified tree traversal.
• Then, GetNextItem, dequeues the
node values from the queue.
void ResetTree(OrderType);
void GetNextItem(ItemType&,
OrderType, bool&);
Revise Tree Class Specification
enum OrderType {PRE_ORDER, IN_ORDER, POST_ORDER};
template<class ItemType>
class TreeType {
public:
// previous member functions
void PreOrder(TreeNode<ItemType>, QueType<ItemType>&)
void InOrder(TreeNode<ItemType>, QueType<ItemType>&)
void PostOrder(TreeNode<ItemType>, QueType<ItemType>&)
private:
TreeNode<ItemType>* root;
QueType<ItemType> preQue;
QueType<ItemType> inQue;
QueType<ItemType> postQue;
};
new private data
new member
functions
template<class ItemType>
void PreOrder(TreeNode<ItemType>tree,
QueType<ItemType>& preQue)
{
if(tree != NULL) {
preQue.Enqueue(tree->info); // “visit”
PreOrder(tree->left, preQue);
PreOrder(tree->right, preQue);
}
}
ResetTree and GetNextItem (cont.)
ResetTree and GetNextItem (cont.)
template<class ItemType>
void InOrder(TreeNode<ItemType>tree,
QueType<ItemType>& inQue)
{
if(tree != NULL) {
InOrder(tree->left, inQue);
inQue.Enqueue(tree->info); // “visit”
InOrder(tree->right, inQue);
}
}
template<class ItemType>
void PostOrder(TreeNode<ItemType>tree,
QueType<ItemType>& postQue)
{
if(tree != NULL) {
PostOrder(tree->left, postQue);
PostOrder(tree->right, postQue);
postQue.Enqueue(tree->info); // “visit”
}
}
ResetTree and GetNextItem (cont.)
ResetTree
template<class ItemType>
void TreeType<ItemType>::ResetTree(OrderType order)
{
switch(order) {
case PRE_ORDER: PreOrder(root, preQue);
break;
case IN_ORDER: InOrder(root, inQue);
break;
case POST_ORDER: PostOrder(root, postQue);
break;
}
}
GetNextItem
template<class ItemType>
void TreeType<ItemType>::GetNextItem(ItemType& item,
OrderType order, bool& finished)
{
finished = false;
switch(order) {
case PRE_ORDER: preQue.Dequeue(item);
if(preQue.IsEmpty())
finished = true;
break;
case IN_ORDER: inQue.Dequeue(item);
if(inQue.IsEmpty())
finished = true;
break;
case POST_ORDER: postQue.Dequeue(item);
if(postQue.IsEmpty())
finished = true;
break;
}
}
Iterative Insertion and Deletion
• Reading Assignment (see textbook)
Comparing Binary Search Trees to Linear Lists
Big-O Comparison
Operation
Binary
Search Tree
Array-
based List
Linked
List
Constructor O(1) O(1) O(1)
Destructor O(N) O(1) O(N)
IsFull O(1) O(1) O(1)
IsEmpty O(1) O(1) O(1)
RetrieveItem O(logN)* O(logN) O(N)
InsertItem O(logN)* O(N) O(N)
DeleteItem O(logN)* O(N) O(N)
*assuming h=O(logN)
Exercises 37-41 (p. 539)
Exercise 17 (p. 537)
Exercise 18 (p. 537)

Weitere ähnliche Inhalte

Ähnlich wie BinarySearchTrees.ppt

1. Add a breadth-first (level-order) traversal function to the binar.pdf
1. Add a breadth-first (level-order) traversal function to the binar.pdf1. Add a breadth-first (level-order) traversal function to the binar.pdf
1. Add a breadth-first (level-order) traversal function to the binar.pdfarjuncp10
 
Fundamentals of data structures
Fundamentals of data structuresFundamentals of data structures
Fundamentals of data structuresNiraj Agarwal
 
Binary Search Tree
Binary Search TreeBinary Search Tree
Binary Search TreeZafar Ayub
 
Cinterviews Binarysearch Tree
Cinterviews Binarysearch TreeCinterviews Binarysearch Tree
Cinterviews Binarysearch Treecinterviews
 
Binary Search Tree
Binary Search TreeBinary Search Tree
Binary Search Treesagar yadav
 
CS-102 BST_27_3_14.pdf
CS-102 BST_27_3_14.pdfCS-102 BST_27_3_14.pdf
CS-102 BST_27_3_14.pdfssuser034ce1
 
Data structures and Algorithm analysis_Lecture4.pptx
Data structures and Algorithm analysis_Lecture4.pptxData structures and Algorithm analysis_Lecture4.pptx
Data structures and Algorithm analysis_Lecture4.pptxAhmedEldesoky24
 
lecture-i-trees.ppt
lecture-i-trees.pptlecture-i-trees.ppt
lecture-i-trees.pptMouDhara1
 
Introduction to data structure by anil dutt
Introduction to data structure by anil duttIntroduction to data structure by anil dutt
Introduction to data structure by anil duttAnil Dutt
 
8 chapter4 trees_bst
8 chapter4 trees_bst8 chapter4 trees_bst
8 chapter4 trees_bstSSE_AndyLi
 
1.1 binary tree
1.1 binary tree1.1 binary tree
1.1 binary treeKrish_ver2
 
1.5 binary search tree
1.5 binary search tree1.5 binary search tree
1.5 binary search treeKrish_ver2
 
nptel 2nd presentation.pptx
nptel 2nd presentation.pptxnptel 2nd presentation.pptx
nptel 2nd presentation.pptxKeshavBandil2
 

Ähnlich wie BinarySearchTrees.ppt (20)

1. Add a breadth-first (level-order) traversal function to the binar.pdf
1. Add a breadth-first (level-order) traversal function to the binar.pdf1. Add a breadth-first (level-order) traversal function to the binar.pdf
1. Add a breadth-first (level-order) traversal function to the binar.pdf
 
Fundamentals of data structures
Fundamentals of data structuresFundamentals of data structures
Fundamentals of data structures
 
Binary Search Tree
Binary Search TreeBinary Search Tree
Binary Search Tree
 
Binary search tree(bst)
Binary search tree(bst)Binary search tree(bst)
Binary search tree(bst)
 
Ch02
Ch02Ch02
Ch02
 
Cinterviews Binarysearch Tree
Cinterviews Binarysearch TreeCinterviews Binarysearch Tree
Cinterviews Binarysearch Tree
 
Binary Search Tree
Binary Search TreeBinary Search Tree
Binary Search Tree
 
CS-102 BST_27_3_14.pdf
CS-102 BST_27_3_14.pdfCS-102 BST_27_3_14.pdf
CS-102 BST_27_3_14.pdf
 
Unit iv data structure-converted
Unit  iv data structure-convertedUnit  iv data structure-converted
Unit iv data structure-converted
 
Data structures and Algorithm analysis_Lecture4.pptx
Data structures and Algorithm analysis_Lecture4.pptxData structures and Algorithm analysis_Lecture4.pptx
Data structures and Algorithm analysis_Lecture4.pptx
 
lecture-i-trees.ppt
lecture-i-trees.pptlecture-i-trees.ppt
lecture-i-trees.ppt
 
Introduction to data structure by anil dutt
Introduction to data structure by anil duttIntroduction to data structure by anil dutt
Introduction to data structure by anil dutt
 
8 chapter4 trees_bst
8 chapter4 trees_bst8 chapter4 trees_bst
8 chapter4 trees_bst
 
1.1 binary tree
1.1 binary tree1.1 binary tree
1.1 binary tree
 
Recursion.ppt
Recursion.pptRecursion.ppt
Recursion.ppt
 
Tree
TreeTree
Tree
 
4.2 bst 03
4.2 bst 034.2 bst 03
4.2 bst 03
 
1.5 binary search tree
1.5 binary search tree1.5 binary search tree
1.5 binary search tree
 
Trees gt(1)
Trees gt(1)Trees gt(1)
Trees gt(1)
 
nptel 2nd presentation.pptx
nptel 2nd presentation.pptxnptel 2nd presentation.pptx
nptel 2nd presentation.pptx
 

Kürzlich hochgeladen

CSR Managerial Round Questions and answers.pptx
CSR Managerial Round Questions and answers.pptxCSR Managerial Round Questions and answers.pptx
CSR Managerial Round Questions and answers.pptxssusera0771e
 
TEST RIG FOR HYDRAULIC FLUID (A3987) - NEOMETRIX DEFENCE
TEST RIG FOR HYDRAULIC FLUID (A3987) - NEOMETRIX DEFENCETEST RIG FOR HYDRAULIC FLUID (A3987) - NEOMETRIX DEFENCE
TEST RIG FOR HYDRAULIC FLUID (A3987) - NEOMETRIX DEFENCENeometrix_Engineering_Pvt_Ltd
 
solar wireless electric vechicle charging system
solar wireless electric vechicle charging systemsolar wireless electric vechicle charging system
solar wireless electric vechicle charging systemgokuldongala
 
Tachyon 100G PCB Performance Attributes and Applications
Tachyon 100G PCB Performance Attributes and ApplicationsTachyon 100G PCB Performance Attributes and Applications
Tachyon 100G PCB Performance Attributes and ApplicationsEpec Engineered Technologies
 
Mastery in Code Review Security Aardwolf Security.pptx
Mastery in Code Review Security Aardwolf Security.pptxMastery in Code Review Security Aardwolf Security.pptx
Mastery in Code Review Security Aardwolf Security.pptxAardwolf Security
 
GENERAL CONDITIONS FOR CONTRACTS OF CIVIL ENGINEERING WORKS
GENERAL CONDITIONS  FOR  CONTRACTS OF CIVIL ENGINEERING WORKS GENERAL CONDITIONS  FOR  CONTRACTS OF CIVIL ENGINEERING WORKS
GENERAL CONDITIONS FOR CONTRACTS OF CIVIL ENGINEERING WORKS Bahzad5
 
Carbohydrates principles of biochemistry
Carbohydrates principles of biochemistryCarbohydrates principles of biochemistry
Carbohydrates principles of biochemistryKomakeTature
 
Oracle_PLSQL_basic_tutorial_with_workon_Exercises.ppt
Oracle_PLSQL_basic_tutorial_with_workon_Exercises.pptOracle_PLSQL_basic_tutorial_with_workon_Exercises.ppt
Oracle_PLSQL_basic_tutorial_with_workon_Exercises.pptDheerajKashnyal
 
Java Presentation on the topic of string
Java Presentation  on the topic of stringJava Presentation  on the topic of string
Java Presentation on the topic of stringRajTangadi
 
Power System electrical and electronics .pptx
Power System electrical and electronics .pptxPower System electrical and electronics .pptx
Power System electrical and electronics .pptxMUKULKUMAR210
 
(Data Communication Series) Zhenbin Li, Zhibo Hu, Cheng Li - SRv6 Network Pro...
(Data Communication Series) Zhenbin Li, Zhibo Hu, Cheng Li - SRv6 Network Pro...(Data Communication Series) Zhenbin Li, Zhibo Hu, Cheng Li - SRv6 Network Pro...
(Data Communication Series) Zhenbin Li, Zhibo Hu, Cheng Li - SRv6 Network Pro...CaoVuThang
 
WATERSHED NOTES IN BOTH HYDROLOGY AND GEOMORPHOLOGY COURSES.pptx
WATERSHED NOTES IN BOTH HYDROLOGY AND GEOMORPHOLOGY COURSES.pptxWATERSHED NOTES IN BOTH HYDROLOGY AND GEOMORPHOLOGY COURSES.pptx
WATERSHED NOTES IN BOTH HYDROLOGY AND GEOMORPHOLOGY COURSES.pptxFormulaMw
 
Modelling Guide for Timber Structures - FPInnovations
Modelling Guide for Timber Structures - FPInnovationsModelling Guide for Timber Structures - FPInnovations
Modelling Guide for Timber Structures - FPInnovationsYusuf Yıldız
 
UNIT4_ESD_wfffffggggggggggggith_ARM.pptx
UNIT4_ESD_wfffffggggggggggggith_ARM.pptxUNIT4_ESD_wfffffggggggggggggith_ARM.pptx
UNIT4_ESD_wfffffggggggggggggith_ARM.pptxrealme6igamerr
 
Java Presentation On Synchronization in Multithreading
Java Presentation On Synchronization in MultithreadingJava Presentation On Synchronization in Multithreading
Java Presentation On Synchronization in MultithreadingRonitGavali1
 
Strategies of Urban Morphologyfor Improving Outdoor Thermal Comfort and Susta...
Strategies of Urban Morphologyfor Improving Outdoor Thermal Comfort and Susta...Strategies of Urban Morphologyfor Improving Outdoor Thermal Comfort and Susta...
Strategies of Urban Morphologyfor Improving Outdoor Thermal Comfort and Susta...amrabdallah9
 
sdfsadopkjpiosufoiasdoifjasldkjfl a asldkjflaskdjflkjsdsdf
sdfsadopkjpiosufoiasdoifjasldkjfl a asldkjflaskdjflkjsdsdfsdfsadopkjpiosufoiasdoifjasldkjfl a asldkjflaskdjflkjsdsdf
sdfsadopkjpiosufoiasdoifjasldkjfl a asldkjflaskdjflkjsdsdfJulia Kaye
 

Kürzlich hochgeladen (20)

CSR Managerial Round Questions and answers.pptx
CSR Managerial Round Questions and answers.pptxCSR Managerial Round Questions and answers.pptx
CSR Managerial Round Questions and answers.pptx
 
Lecture 2 .pptx
Lecture 2                            .pptxLecture 2                            .pptx
Lecture 2 .pptx
 
TEST RIG FOR HYDRAULIC FLUID (A3987) - NEOMETRIX DEFENCE
TEST RIG FOR HYDRAULIC FLUID (A3987) - NEOMETRIX DEFENCETEST RIG FOR HYDRAULIC FLUID (A3987) - NEOMETRIX DEFENCE
TEST RIG FOR HYDRAULIC FLUID (A3987) - NEOMETRIX DEFENCE
 
Lecture 2 .pdf
Lecture 2                           .pdfLecture 2                           .pdf
Lecture 2 .pdf
 
solar wireless electric vechicle charging system
solar wireless electric vechicle charging systemsolar wireless electric vechicle charging system
solar wireless electric vechicle charging system
 
Tachyon 100G PCB Performance Attributes and Applications
Tachyon 100G PCB Performance Attributes and ApplicationsTachyon 100G PCB Performance Attributes and Applications
Tachyon 100G PCB Performance Attributes and Applications
 
Mastery in Code Review Security Aardwolf Security.pptx
Mastery in Code Review Security Aardwolf Security.pptxMastery in Code Review Security Aardwolf Security.pptx
Mastery in Code Review Security Aardwolf Security.pptx
 
GENERAL CONDITIONS FOR CONTRACTS OF CIVIL ENGINEERING WORKS
GENERAL CONDITIONS  FOR  CONTRACTS OF CIVIL ENGINEERING WORKS GENERAL CONDITIONS  FOR  CONTRACTS OF CIVIL ENGINEERING WORKS
GENERAL CONDITIONS FOR CONTRACTS OF CIVIL ENGINEERING WORKS
 
Carbohydrates principles of biochemistry
Carbohydrates principles of biochemistryCarbohydrates principles of biochemistry
Carbohydrates principles of biochemistry
 
Présentation IIRB 2024 Marine Cordonnier.pdf
Présentation IIRB 2024 Marine Cordonnier.pdfPrésentation IIRB 2024 Marine Cordonnier.pdf
Présentation IIRB 2024 Marine Cordonnier.pdf
 
Oracle_PLSQL_basic_tutorial_with_workon_Exercises.ppt
Oracle_PLSQL_basic_tutorial_with_workon_Exercises.pptOracle_PLSQL_basic_tutorial_with_workon_Exercises.ppt
Oracle_PLSQL_basic_tutorial_with_workon_Exercises.ppt
 
Java Presentation on the topic of string
Java Presentation  on the topic of stringJava Presentation  on the topic of string
Java Presentation on the topic of string
 
Power System electrical and electronics .pptx
Power System electrical and electronics .pptxPower System electrical and electronics .pptx
Power System electrical and electronics .pptx
 
(Data Communication Series) Zhenbin Li, Zhibo Hu, Cheng Li - SRv6 Network Pro...
(Data Communication Series) Zhenbin Li, Zhibo Hu, Cheng Li - SRv6 Network Pro...(Data Communication Series) Zhenbin Li, Zhibo Hu, Cheng Li - SRv6 Network Pro...
(Data Communication Series) Zhenbin Li, Zhibo Hu, Cheng Li - SRv6 Network Pro...
 
WATERSHED NOTES IN BOTH HYDROLOGY AND GEOMORPHOLOGY COURSES.pptx
WATERSHED NOTES IN BOTH HYDROLOGY AND GEOMORPHOLOGY COURSES.pptxWATERSHED NOTES IN BOTH HYDROLOGY AND GEOMORPHOLOGY COURSES.pptx
WATERSHED NOTES IN BOTH HYDROLOGY AND GEOMORPHOLOGY COURSES.pptx
 
Modelling Guide for Timber Structures - FPInnovations
Modelling Guide for Timber Structures - FPInnovationsModelling Guide for Timber Structures - FPInnovations
Modelling Guide for Timber Structures - FPInnovations
 
UNIT4_ESD_wfffffggggggggggggith_ARM.pptx
UNIT4_ESD_wfffffggggggggggggith_ARM.pptxUNIT4_ESD_wfffffggggggggggggith_ARM.pptx
UNIT4_ESD_wfffffggggggggggggith_ARM.pptx
 
Java Presentation On Synchronization in Multithreading
Java Presentation On Synchronization in MultithreadingJava Presentation On Synchronization in Multithreading
Java Presentation On Synchronization in Multithreading
 
Strategies of Urban Morphologyfor Improving Outdoor Thermal Comfort and Susta...
Strategies of Urban Morphologyfor Improving Outdoor Thermal Comfort and Susta...Strategies of Urban Morphologyfor Improving Outdoor Thermal Comfort and Susta...
Strategies of Urban Morphologyfor Improving Outdoor Thermal Comfort and Susta...
 
sdfsadopkjpiosufoiasdoifjasldkjfl a asldkjflaskdjflkjsdsdf
sdfsadopkjpiosufoiasdoifjasldkjfl a asldkjflaskdjflkjsdsdfsdfsadopkjpiosufoiasdoifjasldkjfl a asldkjflaskdjflkjsdsdf
sdfsadopkjpiosufoiasdoifjasldkjfl a asldkjflaskdjflkjsdsdf
 

BinarySearchTrees.ppt

  • 1. Binary Search Trees CS 302 – Data Structures Chapter 8
  • 2. What is a binary tree? • Property 1: each node can have up to two successor nodes.
  • 3. • Property 2: a unique path exists from the root to every other node What is a binary tree? (cont.) Not a valid binary tree!
  • 4. Some terminology • The successor nodes of a node are called its children • The predecessor node of a node is called its parent • The "beginning" node is called the root (has no parent) • A node without children is called a leaf
  • 5. Some terminology (cont’d) • Nodes are organize in levels (indexed from 0). • Level (or depth) of a node: number of edges in the path from the root to that node. • Height of a tree h: #levels = L (Warning: some books define h as #levels-1). • Full tree: every node has exactly two children and all the leaves are on the same level. not full!
  • 6. What is the max #nodes at some level l? The max #nodes at level is l 2l 0 2  1 2  2 2  3 2  where l=0,1,2, ...,L-1
  • 7. What is the total #nodes N of a full tree with height h? 0 1 1 1 0 1 1 1 1 0 2 2 ... 2 2 1 ... n h h n n i x x i N x x x x                   using the geometric series: l=0 l=1 l=h-1
  • 8. What is the height h of a full tree with N nodes? 2 1 2 1 log( 1) (log ) h h N N h N O N         
  • 9. Why is h important? • Tree operations (e.g., insert, delete, retrieve etc.) are typically expressed in terms of h. • So, h determines running time!
  • 10. •What is the max height of a tree with N nodes? •What is the min height of a tree with N nodes? N (same as a linked list) log(N+1)
  • 11. How to search a binary tree? (1) Start at the root (2) Search the tree level by level, until you find the element you are searching for or you reach a leaf. Is this better than searching a linked list? No  O(N)
  • 12. Binary Search Trees (BSTs) • Binary Search Tree Property: The value stored at a node is greater than the value stored at its left child and less than the value stored at its right child
  • 13. In a BST, the value stored at the root of a subtree is greater than any value in its left subtree and less than any value in its right subtree! Binary Search Trees (BSTs)
  • 14. Where is the smallest element? Ans: leftmost element Where is the largest element? Ans: rightmost element Binary Search Trees (BSTs)
  • 15. How to search a binary search tree? (1) Start at the root (2) Compare the value of the item you are searching for with the value stored at the root (3) If the values are equal, then item found; otherwise, if it is a leaf node, then not found
  • 16. How to search a binary search tree? (4) If it is less than the value stored at the root, then search the left subtree (5) If it is greater than the value stored at the root, then search the right subtree (6) Repeat steps 2-6 for the root of the subtree chosen in the previous step 4 or 5
  • 17. How to search a binary search tree? Is this better than searching a linked list? Yes !! ---> O(logN)
  • 18. Tree node structure template<class ItemType> struct TreeNode<ItemType> { ItemType info; TreeNode<ItemType>* left; TreeNode<ItemType>* right; };
  • 19. Binary Search Tree Specification #include <fstream.h> struct TreeNode<ItemType>; enum OrderType {PRE_ORDER, IN_ORDER, POST_ORDER}; template<class ItemType> class TreeType { public: TreeType(); ~TreeType(); TreeType(const TreeType<ItemType>&); void operator=(const TreeType<ItemType>&); void MakeEmpty(); bool IsEmpty() const; bool IsFull() const; int NumberOfNodes() const;
  • 20. void RetrieveItem(ItemType&, bool& found); void InsertItem(ItemType); void DeleteItem(ItemType); void ResetTree(OrderType); void GetNextItem(ItemType&, OrderType, bool&); void PrintTree(ofstream&) const; private: TreeNode<ItemType>* root; }; Binary Search Tree Specification (cont.)
  • 21. Function NumberOfNodes • Recursive implementation #nodes in a tree = #nodes in left subtree + #nodes in right subtree + 1 • What is the size factor? Number of nodes in the tree we are examining • What is the base case? The tree is empty • What is the general case? CountNodes(Left(tree)) + CountNodes(Right(tree)) + 1
  • 22. template<class ItemType> int TreeType<ItemType>::NumberOfNodes() const { return CountNodes(root); } template<class ItemType> int CountNodes(TreeNode<ItemType>* tree) { if (tree == NULL) return 0; else return CountNodes(tree->left) + CountNodes(tree->right) + 1; } Function NumberOfNodes (cont.) O(N) Running Time?
  • 24. Function RetrieveItem • What is the size of the problem? Number of nodes in the tree we are examining • What is the base case(s)? 1) When the key is found 2) The tree is empty (key was not found) • What is the general case? Search in the left or right subtrees
  • 25. template <class ItemType> void TreeType<ItemType>:: RetrieveItem(ItemType& item, bool& found) { Retrieve(root, item, found); } template<class ItemType> void Retrieve(TreeNode<ItemType>* tree, ItemType& item, bool& found) { if (tree == NULL) // base case 2 found = false; else if(item < tree->info) Retrieve(tree->left, item, found); else if(item > tree->info) Retrieve(tree->right, item, found); else { // base case 1 item = tree->info; found = true; } } Function RetrieveItem (cont.) O(h) Running Time?
  • 26. Function InsertItem • Use the binary search tree property to insert the new item at the correct place
  • 28. • What is the size of the problem? Number of nodes in the tree we are examining • What is the base case(s)? The tree is empty • What is the general case? Choose the left or right subtree Function InsertItem (cont.)
  • 29. template<class ItemType> void TreeType<ItemType>::InsertItem(ItemType item) { Insert(root, item); } template<class ItemType> void Insert(TreeNode<ItemType>*& tree, ItemType item) { if(tree == NULL) { // base case tree = new TreeNode<ItemType>; tree->right = NULL; tree->left = NULL; tree->info = item; } else if(item < tree->info) Insert(tree->left, item); else Insert(tree->right, item); } Function InsertItem (cont.) O(h) Running Time?
  • 31. • Yes, certain orders might produce very unbalanced trees! Does the order of inserting elements into a tree matter?
  • 33. • Unbalanced trees are not desirable because search time increases! • Advanced tree structures, such as red-black trees, guarantee balanced trees. Does the order of inserting elements into a tree matter? (cont’d)
  • 34. Function DeleteItem • First, find the item; then, delete it • Binary search tree property must be preserved!! • We need to consider three different cases: (1) Deleting a leaf (2) Deleting a node with only one child (3) Deleting a node with two children
  • 36. (2) Deleting a node with only one child
  • 37. (3) Deleting a node with two children
  • 38. • Find predecessor (i.e., rightmost node in the left subtree) • Replace the data of the node to be deleted with predecessor's data • Delete predecessor node (3) Deleting a node with two children (cont.)
  • 39. • What is the size of the problem? Number of nodes in the tree we are examining • What is the base case(s)? Key to be deleted was found • What is the general case? Choose the left or right subtree Function DeleteItem (cont.)
  • 40. template<class ItemType> void TreeType<ItmeType>::DeleteItem(ItemType item) { Delete(root, item); } template<class ItemType> void Delete(TreeNode<ItemType>*& tree, ItemType item) { if(item < tree->info) Delete(tree->left, item); else if(item > tree->info) Delete(tree->right, item); else DeleteNode(tree); } Function DeleteItem (cont.)
  • 41. template <class ItemType> void DeleteNode(TreeNode<ItemType>*& tree) { ItemType item; TreeNode<ItemType>* tempPtr; tempPtr = tree; if(tree->left == NULL) { // right child tree = tree->right; delete tempPtr; } else if(tree->right == NULL) { // left child tree = tree->left; delete tempPtr; } else { GetPredecessor(tree->left, item); tree->info = item; Delete(tree->left, item); } } Function DeleteItem (cont.) 2 children 0 children or 1 child 0 children or 1 child
  • 42. template<class ItemType> void GetPredecessor(TreeNode<ItemType>* tree, ItemType& item) { while(tree->right != NULL) tree = tree->right; item = tree->info; } Function DeleteItem (cont.)
  • 43. template<class ItemType> void TreeType<ItmeType>::DeleteItem(ItemType item) { Delete(root, item); } template<class ItemType> void Delete(TreeNode<ItemType>*& tree, ItemType item) { if(item < tree->info) Delete(tree->left, item); else if(item > tree->info) Delete(tree->right, item); else DeleteNode(tree); } Function DeleteItem (cont.) Running Time? O(h)
  • 45. Tree Traversals There are mainly three ways to traverse a tree: 1) Inorder Traversal 2) Postorder Traversal 3) Preorder Traversal
  • 46. 46 Inorder Traversal: A E H J M T Y ‘J’ ‘E’ ‘A’ ‘H’ ‘T’ ‘M’ ‘Y’ tree Visit left subtree first Visit right subtree last Visit second
  • 47. Inorder Traversal • Visit the nodes in the left subtree, then visit the root of the tree, then visit the nodes in the right subtree Inorder(tree) If tree is not NULL Inorder(Left(tree)) Visit Info(tree) Inorder(Right(tree)) Warning: "visit" implies do something with the value at the node (e.g., print, save, update etc.).
  • 48. 48 Preorder Traversal: J E A H T M Y ‘J’ ‘E’ ‘A’ ‘H’ ‘T’ ‘M’ ‘Y’ tree Visit left subtree second Visit right subtree last Visit first
  • 49. Preorder Traversal • Visit the root of the tree first, then visit the nodes in the left subtree, then visit the nodes in the right subtree Preorder(tree) If tree is not NULL Visit Info(tree) Preorder(Left(tree)) Preorder(Right(tree))
  • 50. 50 ‘J’ ‘E’ ‘A’ ‘H’ ‘T’ ‘M’ ‘Y’ tree Visit left subtree first Visit right subtree second Visit last Postorder Traversal: A H E M Y T J
  • 51. Postorder Traversal • Visit the nodes in the left subtree first, then visit the nodes in the right subtree, then visit the root of the tree Postorder(tree) If tree is not NULL Postorder(Left(tree)) Postorder(Right(tree)) Visit Info(tree)
  • 53. Function PrintTree • We use "inorder" to print out the node values. • Keys will be printed out in sorted order. • Hint: binary search could be used for sorting! A D J M Q R T
  • 54. Function PrintTree (cont.) void TreeType::PrintTree(ofstream& outFile) { Print(root, outFile); } template<class ItemType> void Print(TreeNode<ItemType>* tree, ofstream& outFile) { if(tree != NULL) { Print(tree->left, outFile); outFile << tree->info; // “visit” Print(tree->right, outFile); } } to overload “<<“ or “>>”, see: http://www.fredosaurus.com/notes-cpp/oop-friends/overload-io.html
  • 56. Class Destructor How should we delete the nodes of a tree? Use postorder! Delete the tree in a "bottom-up" fashion
  • 57. Class Destructor (cont’d) TreeType::~TreeType() { Destroy(root); } void Destroy(TreeNode<ItemType>*& tree) { if(tree != NULL) { Destroy(tree->left); Destroy(tree->right); delete tree; // “visit” } } postorder
  • 58. Copy Constructor How should we create a copy of a tree? Use preorder!
  • 59. Copy Constructor (cont’d) template<class ItemType> TreeType<ItemType>::TreeType(const TreeType<ItemType>& originalTree) { CopyTree(root, originalTree.root); } template<class ItemType) void CopyTree(TreeNode<ItemType>*& copy, TreeNode<ItemType>* originalTree) { if(originalTree == NULL) copy = NULL; else { copy = new TreeNode<ItemType>; copy->info = originalTree->info; CopyTree(copy->left, originalTree->left); CopyTree(copy->right, originalTree->right); } } preorder // “visit”
  • 60. ResetTree and GetNextItem • User needs to specify the tree traversal order. • For efficiency, ResetTree stores in a queue the results of the specified tree traversal. • Then, GetNextItem, dequeues the node values from the queue. void ResetTree(OrderType); void GetNextItem(ItemType&, OrderType, bool&);
  • 61. Revise Tree Class Specification enum OrderType {PRE_ORDER, IN_ORDER, POST_ORDER}; template<class ItemType> class TreeType { public: // previous member functions void PreOrder(TreeNode<ItemType>, QueType<ItemType>&) void InOrder(TreeNode<ItemType>, QueType<ItemType>&) void PostOrder(TreeNode<ItemType>, QueType<ItemType>&) private: TreeNode<ItemType>* root; QueType<ItemType> preQue; QueType<ItemType> inQue; QueType<ItemType> postQue; }; new private data new member functions
  • 62. template<class ItemType> void PreOrder(TreeNode<ItemType>tree, QueType<ItemType>& preQue) { if(tree != NULL) { preQue.Enqueue(tree->info); // “visit” PreOrder(tree->left, preQue); PreOrder(tree->right, preQue); } } ResetTree and GetNextItem (cont.)
  • 63. ResetTree and GetNextItem (cont.) template<class ItemType> void InOrder(TreeNode<ItemType>tree, QueType<ItemType>& inQue) { if(tree != NULL) { InOrder(tree->left, inQue); inQue.Enqueue(tree->info); // “visit” InOrder(tree->right, inQue); } }
  • 64. template<class ItemType> void PostOrder(TreeNode<ItemType>tree, QueType<ItemType>& postQue) { if(tree != NULL) { PostOrder(tree->left, postQue); PostOrder(tree->right, postQue); postQue.Enqueue(tree->info); // “visit” } } ResetTree and GetNextItem (cont.)
  • 65. ResetTree template<class ItemType> void TreeType<ItemType>::ResetTree(OrderType order) { switch(order) { case PRE_ORDER: PreOrder(root, preQue); break; case IN_ORDER: InOrder(root, inQue); break; case POST_ORDER: PostOrder(root, postQue); break; } }
  • 66. GetNextItem template<class ItemType> void TreeType<ItemType>::GetNextItem(ItemType& item, OrderType order, bool& finished) { finished = false; switch(order) { case PRE_ORDER: preQue.Dequeue(item); if(preQue.IsEmpty()) finished = true; break; case IN_ORDER: inQue.Dequeue(item); if(inQue.IsEmpty()) finished = true; break; case POST_ORDER: postQue.Dequeue(item); if(postQue.IsEmpty()) finished = true; break; } }
  • 67. Iterative Insertion and Deletion • Reading Assignment (see textbook)
  • 68. Comparing Binary Search Trees to Linear Lists Big-O Comparison Operation Binary Search Tree Array- based List Linked List Constructor O(1) O(1) O(1) Destructor O(N) O(1) O(N) IsFull O(1) O(1) O(1) IsEmpty O(1) O(1) O(1) RetrieveItem O(logN)* O(logN) O(N) InsertItem O(logN)* O(N) O(N) DeleteItem O(logN)* O(N) O(N) *assuming h=O(logN)